12a
0720
(K12a
0720
)
A knot diagram
1
Linearized knot diagam
3 8 9 10 11 12 2 1 7 5 6 4
Solving Sequence
5,11
6 12 7 10 4 1 9 3 8 2
c
5
c
11
c
6
c
10
c
4
c
12
c
9
c
3
c
8
c
2
c
1
, c
7
Ideals for irreducible components
2
of X
par
I
u
1
= hu
56
+ u
55
+ ··· + 2u
2
− 1i
* 1 irreducible components of dim
C
= 0, with total 56 representations.
1
The image of knot diagram is generated by the software “Draw programme” developed by An-
drew Bartholomew(http://www.layer8.co.uk/maths/draw/index.htm#Running-draw), where we modi-
fied some parts for our purpose(https://github.com/CATsTAILs/LinksPainter).
2
All coefficients of polynomials are rational numbers. But the coefficients are sometimes approximated
in decimal forms when there is not enough margin.
1
I. I
u
1
= hu
56
+ u
55
+ · · · + 2u
2
− 1i
(i) Arc colorings
a
5
=
1
0
a
11
=
0
u
a
6
=
1
−u
2
a
12
=
u
−u
3
+ u
a
7
=
−u
2
+ 1
u
4
− 2u
2
a
10
=
−u
u
a
4
=
−u
2
+ 1
u
2
a
1
=
−u
7
+ 4u
5
− 4u
3
+ 2u
u
7
− 3u
5
+ u
a
9
=
u
7
− 4u
5
+ 4u
3
− 2u
−u
9
+ 5u
7
− 7u
5
+ 2u
3
+ u
a
3
=
u
18
− 11u
16
+ 48u
14
− 107u
12
+ 133u
10
− 95u
8
+ 34u
6
− 2u
4
− 3u
2
+ 1
−u
20
+ 12u
18
+ ··· + 5u
4
+ 2u
2
a
8
=
u
23
− 14u
21
+ ··· − 4u
3
− 2u
−u
23
+ 13u
21
+ ··· − 2u
3
+ u
a
2
=
u
45
− 28u
43
+ ··· + 6u
3
+ u
−u
47
+ 29u
45
+ ··· − 2u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
52
+ 132u
50
+ ··· + 12u + 6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
56
+ 27u
55
+ ··· + 4u + 1
c
2
, c
7
u
56
+ u
55
+ ··· + 2u
2
− 1
c
3
u
56
− u
55
+ ··· − 10u − 1
c
4
, c
5
, c
6
c
10
, c
11
u
56
+ u
55
+ ··· + 2u
2
− 1
c
8
u
56
+ 3u
55
+ ··· − 168u − 11
c
9
, c
12
u
56
+ 5u
55
+ ··· + 180u + 41
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
56
+ 5y
55
+ ··· + 20y + 1
c
2
, c
7
y
56
− 27y
55
+ ··· − 4y + 1
c
3
y
56
− 3y
55
+ ··· − 132y + 1
c
4
, c
5
, c
6
c
10
, c
11
y
56
− 71y
55
+ ··· − 4y + 1
c
8
y
56
+ 17y
55
+ ··· − 34956y + 121
c
9
, c
12
y
56
+ 33y
55
+ ··· − 26332y + 1681
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.000570 + 0.044415I
5.23276 − 0.91933I 15.9080 + 0.I
u = −1.000570 − 0.044415I
5.23276 + 0.91933I 15.9080 + 0.I
u = 0.911585 + 0.375481I
−1.58449 + 11.37680I 6.00000 − 9.66829I
u = 0.911585 − 0.375481I
−1.58449 − 11.37680I 6.00000 + 9.66829I
u = 1.012670 + 0.088406I
3.47067 + 5.61232I 12.19644 − 6.38197I
u = 1.012670 − 0.088406I
3.47067 − 5.61232I 12.19644 + 6.38197I
u = −0.905678 + 0.361326I
0.84033 − 6.42701I 10.11884 + 6.13576I
u = −0.905678 − 0.361326I
0.84033 + 6.42701I 10.11884 − 6.13576I
u = −0.904513 + 0.305928I
2.52237 − 4.64171I 11.91273 + 7.20822I
u = −0.904513 − 0.305928I
2.52237 + 4.64171I 11.91273 − 7.20822I
u = 0.878496 + 0.372437I
−3.56192 + 3.49701I 3.83737 − 3.80502I
u = 0.878496 − 0.372437I
−3.56192 − 3.49701I 3.83737 + 3.80502I
u = 0.898756 + 0.245046I
1.83352 + 0.17402I 10.78063 − 0.76974I
u = 0.898756 − 0.245046I
1.83352 − 0.17402I 10.78063 + 0.76974I
u = −0.783787 + 0.371605I
−4.13801 − 3.02974I 2.90976 + 4.77465I
u = −0.783787 − 0.371605I
−4.13801 + 3.02974I 2.90976 − 4.77465I
u = −0.726144 + 0.377443I
−2.67936 + 4.78083I 5.11849 − 1.99129I
u = −0.726144 − 0.377443I
−2.67936 − 4.78083I 5.11849 + 1.99129I
u = 0.738844 + 0.341726I
−0.169675 − 0.099501I 8.43884 − 1.67585I
u = 0.738844 − 0.341726I
−0.169675 + 0.099501I 8.43884 + 1.67585I
u = 0.796382
1.22357 8.30190
u = −0.082668 + 0.594287I
−4.61691 − 8.09385I 1.42454 + 7.00413I
u = −0.082668 − 0.594287I
−4.61691 + 8.09385I 1.42454 − 7.00413I
u = −0.043832 + 0.589687I
−6.36398 − 0.23859I −1.54426 + 0.22557I
u = −0.043832 − 0.589687I
−6.36398 + 0.23859I −1.54426 − 0.22557I
u = 0.076108 + 0.575049I
−2.15243 + 3.25070I 4.40236 − 3.45612I
u = 0.076108 − 0.575049I
−2.15243 − 3.25070I 4.40236 + 3.45612I
u = −0.379547 + 0.324636I
−0.88423 − 4.37253I 5.84758 + 8.42539I
u = −0.379547 − 0.324636I
−0.88423 + 4.37253I 5.84758 − 8.42539I
u = 0.083930 + 0.484869I
−0.48384 + 1.92095I 5.05330 − 5.19671I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.083930 − 0.484869I
−0.48384 − 1.92095I 5.05330 + 5.19671I
u = 0.415514 + 0.176437I
0.918390 + 0.285358I 11.31774 − 2.71319I
u = 0.415514 − 0.176437I
0.918390 − 0.285358I 11.31774 + 2.71319I
u = −0.201404 + 0.389419I
−1.41046 + 1.90845I 3.11125 + 0.73678I
u = −0.201404 − 0.389419I
−1.41046 − 1.90845I 3.11125 − 0.73678I
u = 1.63514 + 0.06538I
5.45645 − 3.31894I 0
u = 1.63514 − 0.06538I
5.45645 + 3.31894I 0
u = −1.64865 + 0.06370I
8.15629 − 1.24210I 0
u = −1.64865 − 0.06370I
8.15629 + 1.24210I 0
u = 1.65013 + 0.08002I
4.30534 + 4.63418I 0
u = 1.65013 − 0.08002I
4.30534 − 4.63418I 0
u = −1.67906 + 0.09461I
5.37634 − 5.28030I 0
u = −1.67906 − 0.09461I
5.37634 + 5.28030I 0
u = −1.68308
10.1668 0
u = −1.68888 + 0.06630I
10.96880 − 1.40460I 0
u = −1.68888 − 0.06630I
10.96880 + 1.40460I 0
u = 1.68815 + 0.09369I
9.93322 + 8.19069I 0
u = 1.68815 − 0.09369I
9.93322 − 8.19069I 0
u = 1.68987 + 0.07840I
11.65850 + 6.13019I 0
u = 1.68987 − 0.07840I
11.65850 − 6.13019I 0
u = −1.68911 + 0.09828I
7.5229 − 13.2200I 0
u = −1.68911 − 0.09828I
7.5229 + 13.2200I 0
u = 1.71055 + 0.00936I
14.8813 + 1.1210I 0
u = 1.71055 − 0.00936I
14.8813 − 1.1210I 0
u = −1.71254 + 0.01833I
13.1646 − 6.0119I 0
u = −1.71254 − 0.01833I
13.1646 + 6.0119I 0
6
II. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
u
56
+ 27u
55
+ ··· + 4u + 1
c
2
, c
7
u
56
+ u
55
+ ··· + 2u
2
− 1
c
3
u
56
− u
55
+ ··· − 10u − 1
c
4
, c
5
, c
6
c
10
, c
11
u
56
+ u
55
+ ··· + 2u
2
− 1
c
8
u
56
+ 3u
55
+ ··· − 168u − 11
c
9
, c
12
u
56
+ 5u
55
+ ··· + 180u + 41
7
III. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
y
56
+ 5y
55
+ ··· + 20y + 1
c
2
, c
7
y
56
− 27y
55
+ ··· − 4y + 1
c
3
y
56
− 3y
55
+ ··· − 132y + 1
c
4
, c
5
, c
6
c
10
, c
11
y
56
− 71y
55
+ ··· − 4y + 1
c
8
y
56
+ 17y
55
+ ··· − 34956y + 121
c
9
, c
12
y
56
+ 33y
55
+ ··· − 26332y + 1681
8
